The Dolbeault complex with weights according to normal crossings

In the present paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the -equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local -solution operator which involves only Cauchy’s Integral Formula (in one complex variable) and behaves well for L ^p -forms with weights according to normal crossings.